A banner demanding 'Piu rispetto per il nostro Mister Allegri' ('More respect for our coach Allegri') was hung by Rossoneri supporters at the San Siro following speculation that the 45-year-old could be sacked in the summer.
"I appreciated the banner and the applause that the fans dedicated to me but there was no need," he told the club's website. "I know how much affection there is here.
"I'm the club coach and I'll be criticised and other names will always be linked to the job and it's been a while since the last time."
On the pitch, Milan twice came from behind to sink Catania and regain third place in the standings.
Nicola Legrottaglie headed the visitors in front on the half-hour mark, only for Mathieu Flamini to equalise on the stroke of half-time.
Gonzalo Bergessio restored the Sicilians' lead on 65 minutes, but two goals in four minutes from substitute Giampaolo Pazzini turned the game in the Rossoneri's favour before a late Mario Balotelli penalty sealed matters.
"The team had the strength to turn this result around and it was an important win and now we have to prepare for the match against Torino next week to make sure we hang on to this deserved third place," Allegri said.
"Pazzini is a extraordinary guy and he deserved his two goals. He's played this season despite the knee and we're lucky to have him."